# American Institute of Mathematical Sciences

October  2022, 21(10): 3421-3424. doi: 10.3934/cpaa.2022107

## On the optimal decay rate of the weakly damped wave equation

 Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

* Corresponding author

Received  June 2022 Published  October 2022 Early access  June 2022

We provide a proof via direct energy estimates of the optimal exponential decay rate of the semigroup generated by the weakly damped wave equation.

Citation: Monica Conti, Lorenzo Liverani, Vittorino Pata. On the optimal decay rate of the weakly damped wave equation. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3421-3424. doi: 10.3934/cpaa.2022107
##### References:
 [1] M. Conti, L. Liverani and V. Pata, The MGT-Fourier model in the supercritical case, J. Differ. Equ., 301 (2021), 543-567. [2] F. Dell'Oro and V. Pata, Second order linear evolution equations with general dissipation, Appl. Math. Optim., 83 (2021), 1877-1917. [3] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [4] G. R. Goldstein, J. A. Goldstein and G. Perla Menzala, On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199. [5] G. R. Goldstein, J. A. Goldstein and G. Reyes, Overdamping and energy decay for abstract wave equations with strong damping, Asymptot. Anal., 88 (2014), 217-232. [6] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

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##### References:
 [1] M. Conti, L. Liverani and V. Pata, The MGT-Fourier model in the supercritical case, J. Differ. Equ., 301 (2021), 543-567. [2] F. Dell'Oro and V. Pata, Second order linear evolution equations with general dissipation, Appl. Math. Optim., 83 (2021), 1877-1917. [3] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [4] G. R. Goldstein, J. A. Goldstein and G. Perla Menzala, On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199. [5] G. R. Goldstein, J. A. Goldstein and G. Reyes, Overdamping and energy decay for abstract wave equations with strong damping, Asymptot. Anal., 88 (2014), 217-232. [6] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
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